Skip to main content
Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.1.41
Chapter 12, Problem 12.1.41

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The line segment starting at P(0, 0) and ending at Q(2, 8)

Verified step by step guidance
1
Identify the points P(0, 0) and Q(2, 8) as the start and end points of the line segment. We want parametric equations that describe all points on the line segment between these two points.
Recall that a parametric equation for a line segment from point P(x_0, y_0) to Q(x_1, y_1) can be written as: \(x = x_0 + t(x_1 - x_0)\) \(y = y_0 + t(y_1 - y_0)\), where the parameter \(t\) varies over an interval.
Substitute the coordinates of P and Q into the parametric form: \(x = 0 + t(2 - 0) = 2t\) \(y = 0 + t(8 - 0) = 8t\).
Determine the interval for the parameter \(t\). Since the line segment starts at P when \(t=0\) and ends at Q when \(t=1\), the interval for \(t\) is \(0 \leq t \leq 1\).
Thus, the parametric equations for the line segment are \(x = 2t\), \(y = 8t\), with \(t\) in the interval \([0, 1]\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing representation of curves and line segments in a flexible way.
Recommended video:
Guided course
08:02
Parameterizing Equations

Line Segment Parameterization

To parameterize a line segment between two points P and Q, use a parameter t that varies over an interval, typically [0,1]. The coordinates are given by linear interpolation: x(t) = x_P + t(x_Q - x_P), y(t) = y_P + t(y_Q - y_P), tracing the segment as t moves from 0 to 1.
Recommended video:
Guided course
08:02
Parameterizing Equations

Parameter Interval

The parameter interval defines the range of t values for which the parametric equations describe the curve. For a line segment, choosing t in [0,1] ensures the curve starts at P when t=0 and ends at Q when t=1, covering all points in between.
Recommended video:
Guided course
05:59
Eliminating the Parameter
Related Practice
Textbook Question

85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)


A circular corral of unit radius is enclosed by a fence. A goat is outside the corral and tied to the fence with a rope of length 0≤a ≤ π (see figure). What is the area of the region (outside the corral) that the goat can reach?


Textbook Question

Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)

Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The segment of the parabola y=2x ²−4, where −1≤x≤5

1
views
Textbook Question

93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.


An ellipse centered at (-2, -3) with major and minor axes of lengths 30 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

1
views
Textbook Question

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

1
views
Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The horizontal line segment starting at P(8, 2) and ending at Q(−2, 2)

1
views